Online Appendix Aggregate Implications of Innovation Policy

Andrew Atkeson* Ariel Burstein $^{†}$

June 2018

Contents (Appendix)

C Quantitative analysis 2
C. 1 Impact elasticities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
C. 2 Data and calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
C. 3 Calibration of additional parameters to solve nonlinear transition dynamics 7
C. 4 Solving the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
C. 5 Additional numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

D Discussion of models not nested in our framework 26
E Variations of baseline model 28
E. 1 Occupation choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
E. 2 Goods and labor used as inputs in research . . . . . . . . . . . . . . . . . . . 29
E. 3 Klette and Kortum (2004) specification . . . . . . . . . . . . . . . . . . . . . . 31
E. 4 Alternative specification of incumbent technology to acquire new products 33

C Quantitative analysis

Section C. 1 provides alternative expressions to measure impact elasticities. Section C. 2 provides additional details of the calibration procedure for our analytic elasticities. Section C. 3 describes how we calibrate additional parameters that are required to solve the model nonlinearly. Section C. 4 provides the equations that are used to solve the model (BGP and transition dynamics). Section C. 5 provides additional details for the results of Section 7 .

C. 1 Additional formulas for impact elasticities

By equation (68),

\begin{matrix} (85) & Θ_{e} = \frac{1}{ρ - 1} \frac{(η_{e} - δ_{e} ζ ({\bar{x}}_{c})) {\bar{x}}_{e}}{\exp {({\bar{g}}_{Z})}^{ρ - 1}} \frac{{\bar{Y}}_{r}}{{\bar{x}}_{e}} \end{matrix}

Given the definition of

G

in (28), we have that

\exp {(G ({\bar{x}}_{c}, {\bar{x}}_{m}, {\bar{x}}_{e}))}^{ρ - 1} - \exp {(G ({\bar{x}}_{c}, {\bar{x}}_{m}, 0))}^{ρ - 1} = (η_{e} - δ_{e} ζ ({\bar{x}}_{c})) {\bar{x}}_{e}

which combined with equation (85) implies equation (36). We can also express

Θ_{e}

where we used

{\bar{s}}_{e} = \frac{η_{e} {\bar{x}}_{e}}{\exp {({\bar{g}}_{Z})}^{ρ - 1}}

and

\frac{{avsize}_{c}}{a_{\bar{s} i z e_{e}}} = \frac{ζ ({\bar{x}}_{c})}{η_{e}}

where

a v \bar{s} {\overset{―}{s i z e}}_{c}

denotes the average size of continuing products and avsize denotes the average size of new products acquired by entering firms. Therefore, given a target for the term in square brackets in equation (36), and given data on

\frac{a v \bar{s} i z e_{c}}{a v \bar{s} i z e_{e}}

and

{\bar{s}}_{e}

, we can back out the implied level of

δ_{e}

By equations (66) and (64),

\begin{matrix} (87) & Θ_{c} = [\frac{1 - {\bar{τ}}_{c}}{1 - {\bar{τ}}_{e}}] [\frac{1}{1 - δ_{e} \frac{a v \overset{―}{s i z z_{c}}}{a_{v} \bar{s} i z e_{e}}}] Θ_{e} = [\frac{1 - {\bar{τ}}_{c}}{ρ - 1}] [\frac{1 + \bar{R}}{\exp ({\bar{g}}_{Y})}] \frac{{\bar{i}}_{r}}{{\bar{v}}^{'}} \end{matrix}

where the second equality uses equations (39) and (86). Therefore, the value of

Θ_{c}

is
independent of the extent of business stealing

δ_{e}, δ_{m}

(or, similarly, is independent of the contribution of entrants to productivity growth).

Finally, by equations (67), (64), and (65),

\begin{matrix} (88) & Θ_{m} = [\frac{1 - {\bar{τ}}_{m}}{1 - {\bar{τ}}_{e}}] [\frac{1 - δ_{m} \frac{a v \bar{s} i z e_{c}}{a v \bar{s} i z e_{m}}}{1 - δ_{e} \frac{a v \bar{s} i z e_{m}}{a v \bar{s} i z e_{e}}}] Θ_{e} = [\frac{1 - {\bar{τ}}_{m}}{1 - {\bar{τ}}_{c}}] [1 - δ_{m} \frac{a v \bar{s} \overset{―}{i z e_{c}}}{{avsize}_{m}}] Θ_{c} \end{matrix}

where the second equation uses equation (87).

C. 2 Data and calibration

We set the time period in the model to one year and calibrate the BGP of the model using average data for the period 1990-2014 (whenever possible).

We first describe how we assign values to the parameters determining the elasticities required to evaluate the aggregate implications of a change in innovation policies. We then describe how we parameterize the model to evaluate its full nonlinear dynamics.

List of model parameters to calculate analytic elasticities The parameters and BGP statistics that we need to assign to calculate our elasticities analytically are: the degree of intertemporal knowledge spillovers

ϕ

, the impact elasticity

Θ

, the share of physical capital in costs

α

, the share of production labor in firms' output (inclusive of production subsidies)

\frac{(1 - α)}{μ}

, the share of innovative investment in firms' output (inclusive of production subsidies)

{\bar{i}}_{r}

, the discount factor (which equals the ratio of the output growth rate to the interest rate,

\tilde{β} = \frac{\exp (\bar{g} γ)}{1 + \bar{R}})

, the parameter governing the intertemporal elasticity of substitution

γ

, consumption-to-output ratio

\frac{\bar{C}}{\bar{Y}}

, and the depreciation rate of physical capital

d_{k}

(which is required to calculate the dynamics of the rental rate

R_{k}

). We set

Θ

equal to its upper bound,

Θ_{e}

. To measure

Θ_{e}

, we use expression (36), which requires assigning values to the elasticity of substitution

ρ

, the baseline productivity growth rate

{\bar{g}}_{Z}

(which, given

α

and

{\bar{g}}_{Y}

, requires a value for the growth rate of the labor force

{\bar{g}}_{L}

), the counterfactual growth rate when investment by entrants is set to zero

G ({\bar{x}}_{c}, {\bar{x}}_{m}, 0)

, and the ratio of innovative investments by entrants to total investments

\frac{{\bar{x}}_{e}}{{\bar{Y}}_{r}}

. To measure

\frac{{\bar{x}}_{e}}{{\bar{Y}}_{r}}

we use equation 39 (together with 40), which require measures of dividends relative to output

\bar{d}

, employment shares of entering firms

{\bar{s}}_{e}

, and innovation subsidy rates for investment by entrants

τ_{e}

. We also require initial BGP values for the innovation subsidies for incumbent firms,

τ_{c}

and

τ_{m}

, and the production subsidy (which we set to undo the distortions on capital accumulation arising from markups and the corporate profits tax). In measuring the rental rate of capital and firm dividends, we introduce a corporate profits tax, which we denote
by

τ_{corp}

. This tax is applied to the variable profits of intermediate good firms and the return to physical capital. In what follows, we describe how we assign values to these parameters and targets, which are reported in Table 2.

As described in the text, we consider two values of

ϕ : ϕ = - 1.6

and

ϕ = 0.96

. We set the elasticity of substitution

ρ

to 4 in the CES aggregator in equation (4). Note that this assumption fixes the term

1 / (ρ - 1)

that enters into our impact elasticity formulas. Holding the equilibrium markup

μ

fixed, changes in this parameter

ρ

have no impact on any other implication of our model for the elasticities of aggregate productivity and welfare with respect to changes in aggregate innovative investment other than through this term. Hence, it is clear that our measured elasticities will be larger if we choose a value of

1 < ρ < 4

and smaller if we choose a value of

ρ > 4

. Note that our upper bound for the impact elasticity in equation (37) is independent of

ρ

We fix a BGP consumption interest rate of

\bar{R} = 4 %

. This rate of return is close to that estimated by Poterba (1998) (5.1%), Hall (2003) (4.46%), and McGrattan and Prescott (2005)

(4.1 %

for 1990-2001). We have set it at the low end of these estimates to reflect the decline in real interest rates that has occurred over the past several decades. We set the growth rate of the labor force

{\bar{g}}_{L} = 0.007

corresponding to the annual growth rate of employment in the nonfinancial corporate sector over the period 1990-2014. We set the intertemporal elasticity parameter

γ \to 1

(i.e.,

\log

utility).

^{27}

Output and innovative investment by incumbent firms To measure innovative investment and factor payments, we use data from the integrated macroeconomic accounts and fixed assets tables for the US nonfinancial corporate sector derived from the US national income and product accounts (NIPA).

Since 2013, NIPA includes measures of investments and stocks in intellectual property products going back to 1929. These intellectual property products include research and development, software, and artistic and literary originals. Research and development by the nonfinancial corporate sector accounts for roughly half of these investments.

In our model, total expenditures on innovative investments correspond to output of the research good

P_{r t} Y_{r t}

, which equals the compensation of research labor

W_{t} L_{r t}

. We presume that what is measured in the data is the innovative investment expenditures of incumbent firms

P_{r t} (x_{c t} + x_{m t})

, and infer innovative investment expenditures of entering

firms

P_{r t} x_{e t}

(since we assume that entrants are not in the nonfinancial corporate sector when they make these expenditures).

We measure aggregate firms' output inclusive of the production subsidy in our model,

(1 + τ_{y}) Y_{t}

, as gross value added of the nonfinancial corporate sector less indirect business taxes less measured innovative investment expenditures by incumbent firms. With these adjustments, we obtain

\frac{{\bar{P}}_{r t} ({\bar{x}}_{c} + {\bar{x}}_{m})}{(1 + τ_{y}) {\bar{Y}}_{t}} = 0.061

(the average of this measure for the period 1990-2014). The corresponding BGP growth rate of output of the final consumption good is

{\bar{g}}_{Y} = 0.025

and the discount factor

\tilde{β} = \exp ({\bar{g}}_{Y}) / (1 + \bar{R}) = 0.986

Payments to physical capital and profits In the data, gross value added of the nonfinancial corporate sector is decomposed into indirect business taxes, compensation of employees, and gross operating surplus. We measure compensation of production labor

W_{t} L_{p t}

in our model as compensation of employees in the nonfinancial corporate sector less measured innovative investment expenditures by incumbent firms.

In our model, a portion of gross operating surplus is paid as rental payments to physical capital, a portion is paid as corporate profits taxes (taxes on income and wealth in the NIPA), and a portion is paid to owners of firms as after-tax variable profits less expenditures on innovative investment. To measure rental payments to physical capital,

R_{k} K_{t}

, we impute the rental rate to physical capital using the standard user cost formula implied by our model,

{\bar{R}}_{k} = \frac{\bar{R}}{1 - τ_{c o r p}} + d_{k}

, with a depreciation rate of physical capital

d_{k} = 0.055

and a corporate profits tax

τ_{c o r p} = 0.161

calculated as described below, which imply

{\bar{R}}_{k} = 0.1027

. We measure

K_{t}

by the replacement value of physical capital ("Nonfinancial assets, nonfinancial corporate business") minus the replacement value of intangible capital ("Nonresidential intellectual property products, nonfinancial corporate business, current cost basis"), which results in

{\bar{K}}_{t} = 2.125 \times (1 + τ_{y}) {\bar{Y}}_{t}

. We measure the depreciation rate

d_{k}

as the ratio of "Consumption of fixed capital, structures, equipment, and intellectual property products, nonfinancial corporate business" minus consumption of intangible capital, to the replacement value of physical capital. In order to measure consumption of intangible capital, we multiply its replacement value by its depreciation rate. We measure the corporate profits tax

τ_{c o r p}

as the ratio of payments of taxes on income and wealth in the data,

0.037 \times (1 + τ_{y}) {\bar{Y}}_{t}

, to the tax base of this tax in our model (i.e., variable profits plus physical capital income net of depreciation,

(1 + τ_{y}) {\bar{Y}}_{t} - {\bar{W}}_{t} {\bar{L}}_{p t} - d_{k} {\bar{K}}_{t} = 0.228 \times (1 + τ_{y}) {\bar{Y}}_{t})

. In choosing this tax base, we have assumed that firms cannot expense their investments in innovation from profits for tax purposes. We make this assumption so that it is possible for the equilibrium to be conditionally efficient in the presence of uniform innovation subsidies and a positive corporate
profits tax.
Following this procedure, we obtain an average share (over the period 1990-2014) of production labor in firms' output (inclusive of production subsidies) of

\frac{{\bar{W}}_{t} {\bar{L}}_{p t}}{(1 + {\bar{τ}}_{y}) {\bar{Y}}_{t}} =

\frac{(1 - α)}{μ} = 0.654

, a share of capital

\frac{{\bar{R}}_{t} {\bar{K}}_{t}}{(1 + {\bar{τ}}_{y}) {\bar{Y}}_{t}} = \frac{α}{μ} = 0.218

, and a share of variable profits in output

(1 - \frac{{\bar{R}}_{k t} {\bar{K}}_{t} + {\bar{W}}_{t} {\bar{L}}_{p t}}{(1 + {\bar{τ}}_{y}) {\bar{Y}}_{t}}) = (\frac{μ - 1}{μ}) = 0.127

, so

α = 0.25

and

μ = 1.146

. We assume that in the initial BGP innovative subsidies are uniform, with

{\bar{τ}}_{e} = {\bar{τ}}_{c} = {\bar{τ}}_{m} = 0.03

, obtained from Tyson and Linden (2012) (Table 7 in that paper reports estimates of corporate R&D tax credit claims relative to R&D business spending in the United States). This gives us

\bar{d} = \frac{{\bar{D}}_{t}}{(1 + τ_{y}) {\bar{Y}}_{t}} = (\frac{μ - 1}{μ}) (1 - {\bar{τ}}_{corp}) - (1 - {\bar{τ}}_{e}) \frac{{\bar{P}}_{r t} ({\bar{x}}_{c} + {\bar{x}}_{m})}{(1 + {\bar{τ}}_{y}) {\bar{Y}}_{t}} = 0.047

In order to remove the distortions induced by the markup and the corporate profits tax in the allocation of physical capital when calculating welfare, we set the production subsidy

{\bar{τ}}_{y} = \frac{μ}{\bar{R} + d_{k}} (\frac{\bar{R}}{(1 - τ_{corp})} + d_{k}) - 1 = 0.24 .^{28}

Our choice of

{\bar{τ}}_{y}

only affects our welfare calculations. Given measures of

α, {\bar{g}}_{L}

, and

{\bar{g}}_{Y}

, we construct estimates of the BGP growth rate of total factor productivity

{\bar{g}}_{Z} = 0.0136

. We calculate the ratio of consumption to output as

\frac{\bar{C}}{\bar{Y}} = 1 - \frac{\bar{I}}{\bar{K}} \bar{K} \overset{―}{\bar{Y}} = 1 - (\exp ({\bar{g}}_{Y}) - (1 - d_{k})) \frac{α (1 + τ_{y})}{μ {\bar{R}}_{k}} = 0.789

Product-level dynamics To measure the employment shares of entering firms

{\bar{s}}_{e}

, we use annual data from the Longitudinal Business Database (LBD) in the United States on the dynamics of establishments and firms that own them. We make the identifying assumption that an establishment in the LBD data corresponds to an intermediate good in the model. Products in entering firms in the model correspond to establishments in the data that are new (they are active in year

t

but not in year

t - 1

) and are owned by new firms. The average value of

{\bar{s}}_{e}

in the period 1990-2014 is 0.027 . If we assume time periods that are longer than one year, entrants represent a higher share in the total number of establishments and in employment. When we solve the model nonlinearly, we calibrate other parameter values using additional information on product-level dynamics.

By equation (40), with

\bar{d} = 0.047

\exp ({\bar{g}}_{Y}) / (1 + \bar{R}) = 0.986

, and

{\bar{s}}_{e} = 0.027

, we obtain

\bar{v} = 1.15

. By equation (39), we obtain

\frac{{\bar{P}}_{r t} {\bar{x}}_{e}}{(1 + τ_{y}) {\bar{Y}}_{t}} = 0.031

, so

{\bar{i}}_{r} = \frac{{\bar{P}}_{r t} ({\bar{x}}_{e} + {\bar{x}}_{c} + {\bar{x}}_{m})}{(1 + τ_{y}) {\bar{Y}}_{t}} =

0.061 + 0.031 = 0.093

. The fiscal cost on innovation spending,

E / Y

, in the initial BGP is given by

0.03 \times {\bar{i}}_{r} = 0.0028

. The ratio

{\bar{x}}_{e} / {\bar{Y}}_{r}

, which is required to calculate

Θ_{e}

in equation (36), is equal to

0.032 / 0.093 = 0.34

Business stealing Based on the derivation of equation (86) in Appendix B, the expression in square brackets in equation (36) can be written as

\frac{\exp {({\bar{g}}_{Z})}^{ρ - 1} - \exp {(G ({\bar{x}}_{c}, {\bar{x}}_{m}, 0))}^{ρ - 1}}{\exp {({\bar{g}}_{Z})}^{ρ - 1}} = (1 - δ_{e} \frac{a v . \bar{s} i z e_{c}}{a v . \overset{―}{s i z e_{e}}}) {\bar{s}}_{e}

We consider two parameterizations of the extent of business stealing for entering firms. First, we set

δ_{e} = 0

, so that the expression in square brackets in equation (36) is equal to

{\bar{s}}_{e} = 0.027

. Second, we let

δ_{e} > 0

, and choose

G ({\bar{x}}_{c}, {\bar{x}}_{m}, 0)

so that

({\bar{g}}_{Z} - G ({\bar{x}}_{c}, {\bar{x}}_{m}, 0)) / {\bar{g}}_{Z}) =

0.257 , which corresponds to estimates of the portion of annual trend productivity growth due to entry in Akcigit and Kerr (2018). Given our estimate of

{\bar{g}}_{Z} = 0.014

and

ρ = 4

, we obtain

G ({\bar{x}}_{c}, {\bar{x}}_{m}, 0) = 0.01

and the expression in square brackets in equation (36) is equal to 0.011. Given our measure

\frac{a v s \overset{―}{z i z} e_{c}}{a v \overset{―}{s i z} e_{e}} = 3

discussed below, our parameterization implies

δ_{e} = 0.20

C. 3 Calibration of additional parameters to solve nonlinear transition dynamics

In the previous section, we described how we assign values to the parameters determining the elasticities required to evaluate the aggregate implications of a change in innovation policies. We now discuss how we assign remaining parameter values to solve for all BGP variables and to solve the transition dynamics of the model nonlinearly and globally (in which we do not impose that

Θ

is at its upper bound of

Θ_{e}

in response to proportional policy changes). These parameter values and targets are reported in Table 3.

In Section C. 2 we described how we choose the value of

δ_{e}

. Here we impose

δ_{m} = δ_{e}

. Moreover, we specify

\begin{matrix} h (x_{m}) = h_{1} x_{m}^{h_{2}} \\ ζ (x_{c}) = ζ_{0} + ζ_{1} x_{c}^{ζ_{2}} \end{matrix}

where

h_{1} > 0, ζ_{1} > 0, 0 < h_{2} < 1

, and

0 < ζ_{2} < 1

. The parameters that need to be assigned are

{δ_{0}, η_{e}, η_{m}, h_{1}, h_{2}, ζ_{0}, ζ_{1}, ζ_{2}}

Suppose we have data on the growth rate of the measure of products

g_{M t} = \log (M_{t + 1} / M_{t})

, as well as data on the fraction of products that are continuing products in incumbent firms

Shares in firms' output, $Y \times (1 + τ_{y})$		Parameters
Production labor compensation, $W L_{p}$	0.655	Elasticity of substitution, $ρ$	4
Physical capital compensation, $R_{k} K$	0.218	Markup, $μ$	1.146
Variable profits, $Π = (1 + τ_{y}) Y - W L_{p} - R_{k} K$	0.127	Physical capital share in costs, $α$	0.25
Innovation investments by incumbents, $P_{r} (x_{c} + x_{m})$	0.061	Intertemporal spillovers, $ϕ$	-1.6 or 0.99
Dividends, $(1 - τ_{corr}) Π - P_{r} ((1 - τ_{c}) x_{c} + (1 - τ_{m}) x_{m})$	0.047	Capital depreciation rate, $d_{k}$	0.055
Innovation investments by entrants, $P_{r} x_{e}$	0.031	Interest rate, $\bar{R}$	0.04
Innovation intensity, $P_{r} (x_{c} + x_{m} + x_{e}) = P_{r} Y_{r}$	0.093	Intertemporal substitution, $γ$	1
Implied allocation of labor, $L_{p} / L$	0.876	Labor growth rate, ${\bar{g}}_{L}$	0.007
		Scientific progress growth, ${\bar{δ}}_{A_{r}}$	0.029 or -0.006
		Business stealing by entrants, $δ_{e}$	0 or 0.20
Policies in initial BGP		Other targets
Innovation subsidies, $τ_{c} = τ_{m} = τ_{e}$	0.03	Output growth rate, ${\bar{g}}_{Y}$	0.025
Output subsidy, $τ_{y}$	0.24	Employment share of entrants, ${\bar{s}}_{e}$	0.027
Corporate profit tax rate, $τ_{corp}$	0.16		0.257
Other model outcomes
Aggregate productivity growth, ${\bar{g}}_{Z}$	0.014	Value incumbents relative to $\bar{Y} (1 + τ_{y})$	1.16
Discount factor, $\tilde{β}$	0.986	Term $\frac{\exp (\overset{―}{\bar{z} z})^{ρ - 1} - \exp {(G ({\bar{x}}_{c}, {\bar{x}}^{2}, 0))}^{ρ - 1}}{\exp (\bar{z})^{p - 1}}$	0.027 or 0.01
Consumption/output, $\bar{C} / \bar{Y}$	0.789	Impact elasticity entry, $Θ_{e}$	0.026 or 0.010

| Shares in firms' output, $Y \times\left(1+\tau_{y}\right)$ | | Parameters | | | :---: | :---: | :---: | :---: | | Production labor compensation, $W L_{p}$ | 0.655 | Elasticity of substitution, $\rho$ | 4 | | Physical capital compensation, $R_{k} K$ | 0.218 | Markup, $\mu$ | 1.146 | | Variable profits, $\Pi=\left(1+\tau_{y}\right) Y-W L_{p}-R_{k} K$ | 0.127 | Physical capital share in costs, $\alpha$ | 0.25 | | Innovation investments by incumbents, $P_{r}\left(x_{c}+x_{m}\right)$ | 0.061 | Intertemporal spillovers, $\phi$ | -1.6 or 0.99 | | Dividends, $\left(1-\tau_{\text {corr }}\right) \Pi-P_{r}\left(\left(1-\tau_{c}\right) x_{c}+\left(1-\tau_{m}\right) x_{m}\right)$ | 0.047 | Capital depreciation rate, $d_{k}$ | 0.055 | | Innovation investments by entrants, $P_{r} x_{e}$ | 0.031 | Interest rate, $\bar{R}$ | 0.04 | | Innovation intensity, $P_{r}\left(x_{c}+x_{m}+x_{e}\right)=P_{r} Y_{r}$ | 0.093 | Intertemporal substitution, $\gamma$ | 1 | | Implied allocation of labor, $L_{p} / L$ | 0.876 | Labor growth rate, $\bar{g}_{L}$ | 0.007 | | | | Scientific progress growth, $\bar{\delta}_{A_{r}}$ | 0.029 or -0.006 | | | | Business stealing by entrants, $\delta_{e}$ | 0 or 0.20 | | Policies in initial BGP | | Other targets | | | Innovation subsidies, $\tau_{c}=\tau_{m}=\tau_{e}$ | 0.03 | Output growth rate, $\bar{g}_{Y}$ | 0.025 | | Output subsidy, $\tau_{y}$ | 0.24 | Employment share of entrants, $\bar{s}_{e}$ | 0.027 | | Corporate profit tax rate, $\tau_{\text {corp }}$ | 0.16 | ![](https://cdn.mathpix.com/cropped/2024_08_28_d2b2a6121bfaf1583836g-08.jpg?height=65&width=577&top_left_y=1578&top_left_x=1218) | 0.257 | | Other model outcomes | | | | | Aggregate productivity growth, $\bar{g}_{Z}$ | 0.014 | Value incumbents relative to $\bar{Y}\left(1+\tau_{y}\right)$ | 1.16 | | Discount factor, $\tilde{\beta}$ | 0.986 | Term $\frac{\exp (\overline{\bar{z} z})^{\rho-1}-\exp \left(G\left(\bar{x}_{c}, \bar{x}^{2}, 0\right)\right)^{\rho-1}}{\exp (\bar{z})^{p-1}}$ | 0.027 or 0.01 | | Consumption/output, $\bar{C} / \bar{Y}$ | 0.789 | Impact elasticity entry, $\Theta_{e}$ | 0.026 or 0.010 |

Table 2: Baseline parameters, targets, and outcomes

f_{c t + 1}

, the fraction of products that are new to incumbent firms measured as the sum of those that are new to society and stolen

f_{m t + 1}

, and the fraction of products that are produced in entering firms measured as the sum of those that are new to society and stolen

f_{e t + 1} = 1 - f_{c t + 1} - f_{m t + 1}

. Suppose we also have data on the aggregate size of continuing products in incumbent firms

s_{c t + 1}

, the aggregate size of products that are new to incumbent firms measured as the sum of those that are new products and those that are stolen

s_{m t + 1}

, and the aggregate size of products that are new to entering firms measured as the sum of those that are new products and those that are stolen

s_{e t + 1} = 1 - s_{c t + 1} - s_{m t + 1}

. The average size of continuing products in continuing firms, new products in incumbent firms, and new products in entering firms is denoted by avsize

_{c t + 1} = \frac{s_{c t + 1}}{f_{c t + 1}}, a v s i z e_{m t + 1} =

\frac{s_{m t + 1}}{f_{m t + 1}}

and avsize

_{e t + 1} = \frac{s_{e t + 1}}{f_{e t + 1}}

, respectively. Time averages of these variables are denoted with a bar.

We calibrate

δ_{0}, η_{e}, η_{m}

and the initial BGP values of

h ({\bar{x}}_{m}), ζ ({\bar{x}}_{c}), {\bar{x}}_{e}

to satisfy the following equations:

\begin{aligned} h ({\bar{x}}_{m}) = {\bar{f}}_{m} \exp ({\bar{g}}_{M}) \\ {\bar{x}}_{e} = {\bar{f}}_{e} \exp ({\bar{g}}_{M}) \\ ζ ({\bar{x}}_{c}) = {avs}^{-} {\overset{―}{i z e}}_{c} \frac{\exp ((ρ - 1) {\bar{g}}_{Z})}{\exp ({\bar{g}}_{M})} \\ η_{m} = {avsize}_{m} \frac{\exp ((ρ - 1) {\bar{g}}_{Z})}{\exp ({\bar{g}}_{M})} \\ η_{e} = {avsize}_{e} \frac{\exp ((ρ - 1) {\bar{g}}_{Z})}{\exp ({\bar{g}}_{M})} \\ δ_{0} = 1 - δ_{m} h ({\bar{x}}_{m}) - δ_{e} {\bar{x}}_{e} - {\bar{f}}_{c} \exp ({\bar{g}}_{M}) . \end{aligned}

Given initial policy ratios,

(\frac{1 - {\bar{τ}}_{c}}{1 - {\bar{τ}}_{e}})

and

(\frac{1 - {\bar{τ}}_{c}}{1 - {\bar{τ}}_{m}})

, we calibrate

ζ^{'} ({\bar{x}}_{c})

and

h^{'} ({\bar{x}}_{m})

using equations (64) and (65).

In order to calibrate the parameters

h_{1}, h_{2}, ζ_{1}

, and

ζ_{2}

, we must know the values of

{\bar{x}}_{c}

and

{\bar{x}}_{m}

. Note that our calibration procedure uses as an input a measure of

({\bar{x}}_{c} + {\bar{x}}_{m}) / {\bar{Y}}_{r}

and implies a value of

{\bar{x}}_{e} / {\bar{Y}}_{r}

, but does not pin down

{\bar{x}}_{c}

and

{\bar{x}}_{m}

separately. To determine the value of

{\bar{x}}_{c}

, we use the following logic. The contribution of investment in acquiring products each period to firm value must be nonnegative. That is, on a BGP, we must have

\bar{v}

at least as large as the value that the firm would obtain if it were to set investment into acquiring new products equal to zero in every period. Given the assumption that

h (0) = 0

, this alternative value of incumbent firms on a BGP is given by

\tilde{v} = {[1 - \frac{\exp ({\bar{g}}_{Y})}{1 + \bar{R}} {\bar{s}}_{c}]}^{- 1} [(1 - τ_{c o r p}) (\frac{μ - 1}{μ}) - (1 - {\bar{τ}}_{c}) \frac{{\bar{P}}_{r t} {\bar{x}}_{c}}{(1 + {\bar{τ}}_{y}) {\bar{Y}}_{t}}]

τ_{c} = τ_{m}

, then

\bar{v} = {[1 - \frac{\exp ({\bar{g}}_{Y})}{1 + \bar{R}} ({\bar{s}}_{c} + {\bar{s}}_{m})]}^{- 1} [(1 - τ_{c o r p}) (\frac{μ - 1}{μ}) - (1 - {\bar{τ}}_{c}) \frac{{\bar{P}}_{r t} ({\bar{x}}_{c} + {\bar{x}}_{m})}{(1 + {\bar{τ}}_{y}) {\bar{Y}}_{t}}]

The requirement that

\tilde{v} \leq \bar{v}

implies that the research expenditures of incumbents on improving continuing products relative to value added must lie between the bounds

\begin{matrix} (89) & \frac{{\bar{P}}_{r t} ({\bar{x}}_{c} + {\bar{x}}_{m})}{(1 + {\bar{τ}}_{y}) {\bar{Y}}_{t}} \geq \frac{{\bar{P}}_{r t} {\bar{x}}_{c}}{(1 + {\bar{τ}}_{y}) {\bar{Y}}_{t}} \geq \\ \frac{1 - \frac{\exp ({\bar{g}}_{Y})}{1 + \bar{R}} {\bar{s}}_{c}}{1 - \frac{\exp (\bar{g} Y)}{1 + \bar{R}} (1 - {\bar{s}}_{e})} \frac{{\bar{P}}_{r t} ({\bar{x}}_{c} + {\bar{x}}_{m})}{(1 + {\bar{τ}}_{y}) {\bar{Y}}_{t}} - \frac{\frac{\exp ({\bar{g}}_{Y})}{1 + \bar{R}} {\bar{s}}_{m}}{1 - \frac{\exp ({\bar{g}}_{Y})}{1 + \bar{R}} (1 - {\bar{s}}_{e})} \frac{(1 - τ_{c o r p})}{(1 - {\bar{τ}}_{c})} (\frac{μ - 1}{μ}) \end{matrix}

In our calibration, we set

\frac{{\bar{P}}_{r} + {\bar{x}}_{c}}{(1 + {\bar{τ}}_{y}) {\bar{t}}_{t}}

in the middle point between the two bounds. Given values of

{\bar{x}}_{m}, h ({\bar{x}}_{m})

, and

h^{'} ({\bar{x}}_{m})

, we determine the values of

h_{0}

and

h_{1}

. Given values of

{\bar{x}}_{c}, ζ ({\bar{x}}_{c})

, and

ζ^{'} ({\bar{x}}_{c})

(which are assigned as described above, independently of

ζ_{2}

) and a value of

0 < ζ_{2} < 1

, we determine the values of

ζ_{0}

and

ζ_{1}

. For the policy exercises in which we do not set

Θ

at its upper bound of

Θ_{e}

(which is independent of

ζ_{2}

), we set

ζ_{2}

halfway between its two bounds, that is,

ζ_{2} = 0.5

. This assumed value of

ζ_{2}

implies that, with business stealing,

Θ = 0.0093 < 0.0102 = Θ_{e}

when considering proportional policy changes.

^{29}

We use LBD data to measure the necessary statistics to implement this calibration procedure. Specifically, we measure

{\bar{g}}_{M} = 0.01

as the annual growth rate in the number of establishments (averaged between 1990 and 2014),

{\bar{f}}_{e} = 0.0775

as the fraction of establishments in the data that are new and are owned by new firms,

{\bar{f}}_{m} = 0.0227

as the fraction of establishments that are new and are owned by firms that are not new, and

{\bar{f}}_{c} = 0.9

as the fraction of establishments that are not new. The employment shares of these three categories of establishments are

{\bar{s}}_{e} = 0.027, {\bar{s}}_{m} = 0.025

, and

{\bar{s}}_{c} = 0.948

. Our results are very similar if we calibrate the model using the statistics implied in Garcia-Macia et al. (2016).

Firm dynamics targets

Growth rate of measure of products,

{\bar{g}}_{M}

Fraction of products in entering firms,

{\bar{f}}_{e}

Fraction of new products in incumbent firms,

{\bar{f}}_{m}

Fraction of continuing products in incumbent firms,

{\bar{f}}_{c} 0.900

Aggr. size of products in entering firms,

{\bar{s}}_{e}

Aggr. size of new products in incumbent firms,

{\bar{s}}_{m} 0.025

Aggr. size of continuing products in incumbent firms,

{\bar{s}}_{c} 0.948

Parameters

Exogenous exit rate, $δ_{0}$	0.091 or 0.071
Productivity step new products by entrants, $η_{e}$	0.358
Productivity step new products by incumbents, $η_{m}$	1.139
Innovation function new products by incumbents, $h_{0}$ and $h_{1}$	0.120 and 0.5
Innovation function products improvement by incumbents, $ζ_{0}, ζ_{1}$ and $ζ_{2}$	$0.995, 0.268$ and 0.5

Other model outcomes

Lower bound

{\bar{P}}_{r t} {\bar{x}}_{c} / ((1 + {\bar{τ}}_{y}) {\bar{Y}}_{t})

0.032

{\bar{P}}_{r t} {\bar{x}}_{c} / ((1 + {\bar{τ}}_{y}) {\bar{Y}}_{t}) 0.047

{\bar{x}}_{m} 0.036

{\bar{x}}_{c} 0.116

h ({\bar{x}}_{m}) 0.023

h^{'} ({\bar{x}}_{m}) 0.314

ζ ({\bar{x}}_{c}) 1.087

ζ^{'} ({\bar{x}}_{c}) 0.393

Table 3: Parameters and targets for non-linear transition dynamics

C. 4 Solving the model

Given parameter values assigned as described above, here we provide a summary of the steps to solve the model's BGP and transition dynamics assuming that the equilibrium allocations are interior (one needs to verify that the conjectured equilibrium that is obtained is interior).

^{30}

BGP The economy starts on an original BGP corresponding to policies

{\bar{τ}}_{c}, {\bar{τ}}_{m}, {\bar{τ}}_{e}

, and

{\bar{τ}}_{y}

. We normalize the level of scientific progress to be

A_{r t} = 1

t = 0

. This gives

A_{r t} =

\exp (t {\bar{g}}_{A r})

. We normalize the population to be

L_{t} = 1

t = 0

. This gives

L_{t} = \exp (t {\bar{g}}_{L})

. We take as given parameters and observed growth rates of aggregate output and labor

{\bar{g}}_{Y}

and

g_{L}

Let

l_{p t} = L_{p t} / L_{t}

and

l_{r t} = L_{r t} / L_{t}

. These variables are constant at

{\bar{l}}_{p}

and

{\bar{l}}_{r}

on the initial BGP. Let

k_{t} = K_{t} / \exp (t {\bar{g}}_{Y})

. This variable is constant at

\bar{k}

on the initial BGP. Let

v_{t} = V_{t} / ((1 + {\bar{τ}}_{y}) Y_{t})

and

p_{r t} = P_{r t} / ((1 + {\bar{τ}}_{y}) Y_{t})

. Both of these variables are constant on a BGP at

\bar{v}

and

{\bar{p}}_{r}

. Let

z_{t} = Z_{t} / \exp (t {\bar{g}}_{Z})

where we solve for the BGP value

{\bar{g}}_{Z}

below. Let

\bar{z}

denote the BGP value of this variable. Let

y_{t} = Y_{t} / \exp (t {\bar{g}}_{Y})

where we calibrate the BGP value

{\bar{g}}_{Y}

. Let

\bar{y}

denote the BGP value of this variable. Let

c_{t} = C_{t} / \exp (t {\bar{g}}_{Y})

. Let

\bar{c}

denote the BGP value of this variable. The BGP equations to be solved for the value of the state variables

Z_{0}

and

K_{0}

are as follows.

Growth rates

{\bar{g}}_{Z}, {\bar{g}}_{A r}

\begin{matrix} {\bar{g}}_{Z} = (1 - α) ({\bar{g}}_{Y} - {\bar{g}}_{L}) \\ {\bar{g}}_{A r} = (1 - ϕ) {\bar{g}}_{Z} - {\bar{g}}_{L} \end{matrix}

Interest rate and rental rate of physical capital

\bar{R}

and

{\bar{R}}_{k}

\begin{matrix} 1 + \bar{R} = \exp (γ ({\bar{g}}_{Y} - {\bar{g}}_{L})) / β \\ \bar{R} = (1 - τ_{c o r p}) ({\bar{R}}_{k} - d_{k}) \end{matrix}

Innovative investment

{\bar{x}}_{c}, {\bar{x}}_{m}, {\bar{x}}_{e}

, and

{\bar{Y}}_{r}

\begin{matrix} \frac{1 - {\bar{τ}}_{c}}{1 - {\bar{τ}}_{e}} η_{e} = (1 - δ_{0} - δ_{m} h ({\bar{x}}_{m}) - δ_{e} {\bar{x}}_{e}) ζ^{'} ({\bar{x}}_{c}) \\ \frac{1 - {\bar{τ}}_{m}}{1 - {\bar{τ}}_{e}} η_{e} = η_{m} h^{'} ({\bar{x}}_{m}) \\ {\bar{g}}_{Z} = \frac{1}{ρ - 1} \log ((1 - δ_{0} - δ_{m} h ({\bar{x}}_{m}) - δ_{e} {\bar{x}}_{e}) ζ ({\bar{x}}_{c}) + η_{m} h ({\bar{x}}_{m}) + η_{e} {\bar{x}}_{e}) \\ {\bar{x}}_{c} + {\bar{x}}_{m} + {\bar{x}}_{e} = {\bar{Y}}_{r} \end{matrix}

Employment share of entering firms, value of intangible capital, price of the research good

{\bar{s}}_{e}

\bar{v}

and

{\bar{p}}_{r}

\begin{matrix} {\bar{s}}_{e} = \frac{η_{e} {\bar{x}}_{e}}{\exp ((ρ - 1) {\bar{g}}_{Z})} \\ (90) & \bar{v} = \frac{[(1 - τ_{c o r p}) \frac{μ - 1}{μ} - {\bar{p}}_{r} ((1 - {\bar{τ}}_{c}) {\bar{x}}_{c} + (1 - {\bar{τ}}_{m}) {\bar{x}}_{m})]}{1 - \frac{\exp ({\bar{g}}_{Y})}{1 + \bar{R}} (1 - {\bar{s}}_{e})} \\ (1 - {\bar{τ}}_{e}) {\bar{p}}_{r} {\bar{x}}_{e} = {\bar{s}}_{e} \frac{\exp ({\bar{g}}_{Y})}{1 + \bar{R}} \bar{v} \end{matrix}

Note that by combining the last two expressions, we can rewrite

\bar{v}

\begin{matrix} (91) & \bar{v} = \frac{[(1 - τ_{c o r p}) \frac{μ - 1}{μ} - {\bar{p}}_{r} ((1 - {\bar{τ}}_{c}) {\bar{x}}_{c} + (1 - {\bar{τ}}_{m}) {\bar{x}}_{m} + (1 - {\bar{τ}}_{e}) {\bar{x}}_{e})]}{1 - \frac{\exp ({\bar{g}}_{Y})}{1 + \bar{R}}} \end{matrix}

Innovation intensity and allocation of labor

{\bar{i}}_{r}, {\bar{l}}_{p}, {\bar{l}}_{r}

\begin{matrix} {\bar{i}}_{r} = {\bar{p}}_{r} {\bar{Y}}_{r} \\ \frac{{\bar{l}}_{p}}{{\bar{l}}_{r}} = \frac{1 - α}{μ} \frac{1}{{\bar{i}}_{r}} \\ {\bar{l}}_{p} + {\bar{l}}_{r} = 1 \end{matrix}

Levels of aggregate productivity

\bar{z}

{\bar{Y}}_{r} = {\bar{z}}^{ϕ - 1} {\bar{l}}_{r}

Output, consumption, and capital stock

\bar{y}, \bar{c}

and

\bar{k}

\begin{matrix} {\bar{R}}_{k} = (1 + τ_{y}) \frac{α}{μ} \frac{\bar{y}}{\bar{k}} \\ \bar{y} = {\bar{z}}^{α} {\bar{l}}_{p}^{1 - α} \\ \bar{c} = \bar{y} - (\exp ({\bar{g}}_{Y}) + (1 - d_{k})) \bar{k} \end{matrix}

New BGP We consider a permanent change in innovation policies from

{\bar{τ}}_{c}, {\bar{τ}}_{m}, {\bar{τ}}_{e}

{\bar{τ}}_{c}^{'}, {\bar{τ}}_{m}^{'}, {\bar{τ}}_{e}^{'}

that is unanticipated and starts in period

t = 0

. Thus, the economy starts in period

t = 0

with the state variables

A_{r 0} = L_{0} = 1

, and

{\bar{Z}}_{0}, {\bar{K}}_{0}

solved for above. The economy follows a transition path to a new BGP. We can use the same procedure as above to compute the new BGP, which we denote with primes. The terminal condition we will use when we solve for the transition dynamics is that we end up at the new BGP allocation.

Transition dynamics The two states of the model are

z_{t}

and

k_{t}

. Given

z_{0}, k_{0}

, we wish to solve for a sequence

{z_{t + 1}, k_{t + 1}}_{0}^{T}

, as well as the remaining intra-period variables that satisfy the Euler equations and the terminal conditions that

z_{t}, k_{t}

converges to the new BGP

{\bar{z}}^{'}, {\bar{k}}^{'}

Let

Y_{t}

denote the vector of intra-period variables:

Y_{t} = (x_{c t}, x_{m t}, x_{e t}, Y_{r t}, l_{r t}, l_{p t}, i_{r t}, p_{r t}, y_{t}, c_{t})

We first show how we solve for

Y_{t}

, given

{z_{t}, z_{t + 1}, k_{t}, k_{t + 1}}

. Specifically, we solve for

(x_{c t}, x_{m t}, x_{e t}, Y_{r t}, l_{r t}, l_{p t}, i_{r t}, p_{r t})

given

(z_{t}, z_{t + 1})

. We then solve for

(y_{t}, c_{t})

given

(k_{t}, k_{t + 1})

Allocation of innovative investment

x_{c t}, x_{m t}, x_{e t}, Y_{r t}

(assuming that the allocation is interior):

\begin{matrix} (92) & \log z_{t + 1} - \log z_{t} + {\bar{g}}_{z} = \frac{1}{ρ - 1} \log ((1 - δ_{0} - δ_{m} h (x_{m t}) - δ_{e} x_{e t}) ζ (x_{c t}) + η_{m} h (x_{m t}) + η_{e} x_{e t}) \\ (93) & \frac{1 - {\bar{τ}}_{c}}{1 - {\bar{τ}}_{e}} η_{e} = (1 - δ_{0} - δ_{m} h (x_{m t}) - δ_{e} x_{e t})) ζ^{'} (x_{c t}) \\ (94) & \frac{1 - {\bar{τ}}_{m}}{1 - {\bar{τ}}_{e}} η_{e} = η_{m} h^{'} (x_{m t}) \end{matrix}

\begin{matrix} (95) & x_{c t} + x_{m t} + x_{e t} = Y_{r t} \end{matrix}

Allocation of labor

l_{r t}, l_{p t}

\begin{matrix} (96) & Y_{r t} = z_{t}^{ϕ - 1} l_{r t} \\ (97) & l_{p t} + l_{r t} = 1 \end{matrix}

Price of the research and innovation intensity

i_{r t}, p_{r t} :

\begin{matrix} (98) & \frac{l_{p t}}{l_{r t}} = \frac{1 - α}{μ} \frac{1}{i_{r t}} \\ (99) & i_{r t} = p_{r t} Y_{r t} \end{matrix}

Output and consumption

y_{t}, c_{t}

\begin{matrix} (100) & y_{t} = z_{t} k_{t}^{α} l_{p t}^{1 - α} \\ (101) & c_{t} = y_{t} + (1 - d_{k}) k_{t} - \exp ({\bar{g}}_{Y}) k_{t + 1} \end{matrix}

We now consider the Euler equations that must be satisfied.
Standard physical capital Euler equation:

\begin{aligned} (102) & R_{t} & = (1 - τ_{corp}) ((1 + τ_{y}) \frac{α}{μ} \frac{y_{t + 1}}{k_{t + 1}} - d_{k}) \\ (103) & 1 & = (1 + R_{t}) \tilde{β} \exp (- {\bar{g}}_{Y}) {(\frac{c_{t + 1}}{c_{t}})}^{- γ} \end{aligned}

Value function and zero profit at entry condition:

\begin{aligned} v_{t} = (1 - τ_{c o r p}) \frac{μ - 1}{μ} - p_{r t} ((1 - τ_{c}) x_{c t} + (1 - τ_{m}) x_{m t}) + \\ \frac{\exp ({\bar{g}}_{Y} - (ρ - 1) {\bar{g}}_{Z})}{1 + R_{t}} v_{t + 1} (\frac{y_{t + 1}}{y_{t}}) {(\frac{z_{t}}{z_{t + 1}})}^{ρ - 1} [(1 - δ_{0} - δ_{m} h (x_{m t}) - δ_{e} x_{e t})) ζ (x_{c t}) + η_{m} h (x_{m t})] \\ and \end{aligned}

\begin{matrix} (105) & (1 - τ_{e t}) p_{r t} = η_{e} \frac{\exp ({\bar{g}}_{Y} - (ρ - 1) {\bar{g}}_{Z})}{1 + R_{t}} v_{t + 1} (\frac{y_{t + 1}}{y_{t}}) {(\frac{z_{t}}{z_{t + 1}})}^{ρ - 1} \end{matrix}

Let

X_{t} = (z_{t}, k_{t}, v_{t - 1})

denote the vector of state variables extended with the lagged value of

v_{t}

. If we specify the triple

X_{t}, X_{t + 1}, X_{t + 2}

, we can solve for

Y_{t}

and

Y_{t + 1}

, and we calculate the Euler equations above. We then solve for the path of

{X_{t}, Y_{t}}

using standard linear or nonlinear methods.

Solving transition taking as given a path of innovation intensities or research labor We now consider the problem of finding innovation policies to implement an equilibrium with a pre-specified path of the allocation of labor to research

{l_{r t}}

or, equivalently given equation (98), a path for the innovation intensity of economy

{i_{r t}}

. We begin with an analysis of our simple model in which only entering firms invest in innovation. We then extend the analysis to include investment by incumbent firms. We impose that the sequence

{l_{r t}}

converges to a constant value

{\bar{l}}_{r}^{'}

associated with the new BGP.

Given

z_{0}, k_{0}

, consider a given path of the allocation of research labor

{l_{r t}}

(or, similarly, the innovation intensity of the economy

{i_{r t}})

. Assume that a given level of the tax

τ_{c o r p}

is fixed. We solve for the implied allocation and the sequence of entry subsidies that implement this allocation as an equilibrium as follows.

Equations (92), (95), (96), (97), (98), and (99) are used recursively to construct the implied sequences in the transition for

{z_{t + 1}, x_{e t}, Y_{r t}, l_{p t}, i_{r t}, p_{r t}}

. The equilibrium sequences of consumption, output, physical capital and the real interest rate

c_{t}, y_{t}, k_{t + 1}, R_{t}

in the transition are given as the solutions to equations (100), (101), (102), and (103) together with the terminal condition that the capital stock converge to its new BGP value

lim_{t \to \infty} k_{t + 1} = {\bar{k}}^{'}

. These equations are those equations characterizing equilibrium in the growth model with time-varying exogenous supply of production labor

l_{p t}

and aggregate productivity

z_{t}

Solving equation (104) recursively, the value of a product of size one relative to output

v_{t}

is given by

v_{t} = [d_{t} + \sum_{k = 1}^{\infty} d_{t + k} (\frac{y_{t + k}}{y_{t}}) {(\frac{z_{t}}{z_{t + k}})}^{ρ - 1} \exp {(\bar{g} Y - (ρ - 1) {\bar{g}}_{Z})}^{k} \prod_{j = 0}^{k - 1} \frac{1 - δ_{c t + j}}{1 + R_{t + j}}]

where

1 - δ_{c t} = 1 - δ_{0} - δ_{e} x_{e t}

and

d_{t} = (1 - τ_{c o r p}) \frac{μ - 1}{μ}

. The sequence of innovation policies

{τ_{e t}}

that implement this allocation as an equilibrium is then given from equation (105).

Observe that when we extend this result to the model with innovative investment
by incumbents, we can no longer uniquely identify the feasible allocation corresponding to a given path of the allocation of labor to research

{l_{r t}}

. Instead, we must specify a rule for interior investment by incumbents

x_{c t}

and

x_{m t}

as a function of

Y_{r t}

that allows for

x_{e t}

strictly positive. One such rule would be to specify that

x_{c t} / Y_{r t}

and

x_{m t} / Y_{r t}

are constant fractions that sum to less than one. Then, from equation (95), we can solve for an interior value of

x_{e t}

. With such a rule for allocating innovative investment across firms, we can repeat the procedure for finding innovation policies that implement the implied allocation as an equilibrium as follows.

Again equations (92), (95), (96), (97), (98), and (99) and our rule for allocating innovative investment by incumbents are used recursively to construct the implied sequences in the transition for

{z_{t + 1}, x_{c t}, x_{m t}, x_{e t}, Y_{r t}, l_{p t}, i_{r t}, p_{r t}}

. The equilibrium sequences of consumption, output, physical capital, and the real interest rate

c_{t}, y_{t}, k_{t + 1}, R_{t}

in the transition are given as the solutions to equations (100), (101), (102), and (103) together with the terminal condition that the capital stock converge to its new BGP value

lim_{t \to \infty} k_{t + 1} = {\bar{k}}^{'}

The problem of finding policies to implement this allocation as an equilibrium is somewhat more complex than is the case in the simple model. We can use equations (93) and (94) to find the required values of

(1 - τ_{c t}) / (1 - τ_{e t})

and

(1 - τ_{m t}) / (1 - τ_{e t})

by plugging the allocation into the right-hand side of these equations. We then have to solve for the entire sequence of entry subsidies

{τ_{e t}}

as follows. The dividend to an incumbent product is now

d_{t} = (1 - τ_{c o r p}) \frac{μ - 1}{μ} - (1 - τ_{e t}) p_{r t} (\frac{1 - τ_{c t}}{1 - τ_{e t}} x_{c t} + \frac{1 - τ_{m t}}{1 - τ_{e t}} x_{m t})

The value of a product of size one relative to output

v_{t}

is given as above, using this expression for dividends and with

1 - δ_{c t}

now including business stealing by incumbent firms. The sequence of entry subsidies that implements this allocation as an equilibrium is then given by the solution to equation (105).

Alternatively to specifying interior paths for

x_{c t} / Y_{r t}

and

x_{m t} / Y_{r t}

, we can specify timevarying policy ratios

\frac{1 - τ_{c t}}{1 - τ_{e t}}

and

\frac{1 - τ_{m t}}{1 - τ_{e t}}

, and solve for

x_{c t}, x_{m t}

, and

x_{e t}

using equations (95), (93), and (94). At this point, we must verify that the solution to this system of equations implies allocations of innovative investment that are interior. After solving for a timevarying path of

{τ_{e t}}

as described above, we calculate

{τ_{c t}, τ_{m t}}

given the innovation policy ratios.

C. 5 Additional numerical results

In this section, we report additional details and figures for our innovation policy counterfactuals considered in Section 7. We first consider our baseline proportional policy
changes. We then consider nonproportional policy changes. Finally, we consider proportional policy changes that are accompanied by annual exogenous productivity shocks.

Figures 1 and 2 display, for each of our four specifications, the transition dynamics (solving the model fully nonlinearly) of aggregate productivity and output in the first 100 and 20 years, respectively, after the proportional policy change that increases research labor permanently by

10 %

. Figure 3 displays the 20-year transition dynamics of fiscal expenditures

E / Y

and the innovation subsidy rate

τ_{e t}

that produces the

10 %

permanent increase in research labor. Recall that, with uniform innovation policies, the fiscal cost of these policies in terms of the change in

E / Y

from the initial BGP to the new BGP required to induce a given change in the innovation intensity of the economy is given as in Proposition 4. In Figure 3 we see that the increase in innovation subsidies in the early phase of the transition is smaller than in the new BGP, especially with low intertemporal knowledge spillovers

ϕ

. With low

ϕ

, the price of the research good is expected to rise quickly during the transition, so that (due to intertemporal substitution) innovation subsidies do not have to be as large in order to induce the same increase in total expenditures.

Figure 4 compares the path of aggregate productivity and output calculated using the log-linear approximation (as in Table 1) and the nonlinear solution method. The small differences between the linear and nonlinear solutions (which vanish in the new BGP) result in very small differences in welfare. For example, with high

ϕ

and no business stealing (in which the differences between the solutions in the first 100 years are the largest in Figure 4), the consumption equivalent changes in welfare are very similar:

20.96 %

when solving the model nonlinearly and 20.08% when solving the model via linear approximation.

We next consider nonproportional changes in innovation subsidies, which satisfy restriction (32). Specifically, we solve for the dynamics of innovation subsidies for entrants,

τ_{e t}

, such that research labor increases by

10 %

permanently and both

\frac{1 - τ_{c t}}{1 - τ_{e t}}

and

\frac{1 - τ_{m t}}{1 - τ_{e t}}

fall from 1 to 0.95 . That is, innovation investments by incumbents are disproportionately subsidized relative to innovation investments by entrants.

We first assume that there is no business stealing. Figure 5 displays the transition dynamics of aggregate productivity and output in the first 100 years when solving the model via linearization (around the initial BGP) and nonlinearly. Given that the initial BGP allocation of innovative investments is conditionally efficient, Proposition 5 applies. Up to a first-order approximation, the transition dynamics of aggregate productivity and output only depend on the aggregate change in research labor and are the same for proportional and nonproportional policy changes (so that the transition dynamics coincide with those displayed in Figure 4 under the linearized solution). Figure 5 shows that the difference between the linearized (using the expressions referenced in Proposition 5) and nonlinear
transition dynamics are quite small. With

ϕ = - 1.6

, the resulting change in welfare is

2.6 %

based on the linearized solution and

2.5 %

based on the nonlinearized solution. With

ϕ = 0.96

, the resulting change in welfare is

20.1 %

based on the linearized solution and

20.6 %

based on the nonlinearized solution.

We next assume that there is business stealing. In this case, the initial BGP allocation of innovative investment is not conditionally efficient, so Proposition 5 does not apply. Since policy changes are nonproportional, Proposition 7 does not apply either. However, in this case we can apply Proposition 6, which provides a first-order approximation around the new BGP. Figure 6 displays the transition dynamics of aggregate productivity and output in the first 100 years, in the presence of business stealing, when solving the model via linearization (as indicated in Proposition 6) and nonlinearly. In this case, there are additional output gains from reallocating innovative investment from entrants to incumbent firms (since

Θ_{c}

and

Θ_{m}

are larger than

Θ_{e}

). The welfare gains when solving the model nonlinearly are

5.4 %

with low

ϕ

and

16.2 %

with high

ϕ

(versus

1.7 %

and

6.9 %

, respectively, when considering a proportional policy change). The differences between the linearized and nonlinear transition dynamics are quite modest given the large policy change that we consider. With

ϕ = - 1.6

, the resulting change in welfare is

5.2 %

based on the linearized solution and

5.4 %

based on the nonlinearized solution. With

ϕ = 0.96

, the resulting change in welfare is

14.7 %

based on the linearized solution and

16.2 %

based on the nonlinearized solution.

Our results in Section 7 suggest that it would be hard to verify whether innovation policies yield large output and welfare gains using medium-term data on the response of aggregates to changes in innovation policies. We illustrate this point in Figure 7. In that figure, we show results obtained from simulating the response of aggregates in our model (with and without business stealing, with low and high

ϕ

) to our baseline proportional increase in innovation subsidies in an extended version of our model with Hicks-neutral AR1 productivity shocks with a persistence of 0.9 and an annual standard deviation of 0.018. We introduce these shocks as a proxy for business cycle shocks around the BGP. We show histograms generated from 3,000 simulations of the log-linearized model for the first 20 years of the transition. The units on the horizontal axis show the log of the ratio of detrended output (in the upper panels) and productivity (in the lower panels) at the end of the 20th year of transition to initial output or productivity. The vertical axis shows the frequency of the corresponding outcome for output or productivity. The red bars show results for the model with low intertemporal knowledge spillovers

ϕ

, and the blue bars show the results with high spillovers

ϕ

. We can observe in each panel that the distribution represented by the blue bars is slightly to the right of that represented by
the red bars. But it is also clear in each panel that, using either output or productivity, it would be very hard to distinguish the degree of intertemporal knowledge spillovers (and, hence, the long-term effects from this innovation subsidy) in aggregate time series data even if we had the benefit of a true policy experiment.

Figure 1: 100-year transition dynamics to a 10% permanent increase in research labor via proportional innovation policies, nonlinear solution

Figure 2: 20-year transition dynamics to a 10% permanent increase in research labor via proportional innovation policies, nonlinear solution

Figure 3: Fiscal expenditures and innovation subsidy rate, 20-year transition dynamics to a

10 %

permanent increase in research labor via proportional innovation policies, nonlinear solution

Figure 4: 100-year transition dynamics to a 10% permanent increase in research labor via proportional innovation policies, nonlinear versus linear solution

Figure 5: 100-year transition dynamics to a 10% permanent increase in research labor via nonproportional innovation policies, no business stealing, nonlinear versus linear solution

Figure 6: 100-year transition dynamics to a

10 %

permanent increase in research labor via nonproportional innovation policies, with business stealing, nonlinear versus linear solution

Figure 7: Histogram of 20-year increase in aggregate output and productivity to a permanent

10 %

increase in research labor, including productivity shocks

D Discussion of models not nested in our framework

As discussed in Section 4.4, our model nests several influential models in the literature on firms' innovative investments and aggregate growth. In constructing our model, we rely on three key assumptions in deriving our analytical results: (i) the markup

μ

is constant across products and time, (ii) the costs and benefits of innovative investments by incumbent firms scale with firm size, and (iii) all incumbent firms share the same technologies for innovative investment. Some recent papers in the literature examine models of firms' investments in innovation in which one or more of these key assumptions do not hold. We discuss some of these alternative model specifications here and the extent to which our results may be applied to these models.

Consider first our assumption that markups

μ

are constant across intermediate goods
and time. We rely on this assumption to derive the formula (6) for aggregate productivity that plays a central role in our derivation of our analytical results. If we consider an alternative model in which both productivity indices

z

and markups

μ

on intermediate goods had support on a grid, and if we let

M_{t} (z_{n}, μ_{j})

denote the measure of intermediate goods with productivity index

z_{n}

and markup

μ_{j}

, then aggregate productivity

Z_{t}

would be given by

Z_{t} = \frac{{(\sum_{n} \sum_{j} z_{n}^{ρ - 1} μ_{j}^{1 - ρ} M_{t} (z_{n}, μ_{j}))}^{ρ / (ρ - 1)}}{\sum_{n} \sum_{j} z_{n}^{ρ - 1} μ_{j}^{- ρ} M_{t} (z_{n}, μ_{j})}

In general, then, in a model in which markups vary across intermediate goods, one must keep track of the evolution of the joint distribution of productivity indices

z

and markups

μ

across products to compute the evolution of aggregate productivity.

^{31}

Peters (2016) presents a model that emphasizes the interaction of firms' investments in innovation and their markups. He introduces a Neo-Schumpeterian model in which incumbent firms invest to improve their own products, and entering and incumbent firms invest to acquire new products. An incumbent firm that innovates on its own product charges a higher markup on its product relative to an entering firm or an incumbent firm acquiring a new product. As a result of this assumption, the joint distribution of markups and productivity indices across products varies over time. Hence, there is no simple analog to the equation (28) we derive in Lemma 3 linking aggregate innovative investments by firms and aggregate productivity growth. Instead, one must keep track of the evolution of the measure

M_{t} (z, μ)

to compute the dynamics of aggregate productivity implied by his model. Innovation policy interacts with competition policy in that investments in innovation by entrants and incumbent firms seeking to acquire new products have an additional impact on the growth of aggregate productivity that arises from the impact of this type of innovation on the portion of products sold at a low markup.

Consider next our assumption that the costs and benefits of innovative investments scale with firm size. This assumption is key to our derivation of equation (28) in Lemma 3 linking aggregate innovative investments by firms and aggregate productivity growth. Akcigit and Kerr (2018) estimate an alternative model of firms' investments in innovation and firm dynamics in which investments by incumbent firms in acquiring new products do not scale proportionally with firm size. In general, in a model of this kind, one must

^{31}

If the distribution of markups across products is independent of the productivity index of the product and if the marginal distribution of markups is constant over time, then the terms

\sum_{j} μ_{j}^{1 - ρ} M_{t} (z_{n}, μ_{j})

and

\sum_{j} μ_{j}^{- ρ} M_{t} (z_{n}, μ_{j})

do not vary with

z_{n}

or with

t

. In this case, the formula above for aggregate productivity reduces to our formula (6) times a constant. In this case, we can extend our analytical results to cover a model with markups that vary across intermediate goods.
keep track of the distribution of incumbent firms by size to compute the transition dynamics of aggregate productivity implied by the model. In Appendix E.4, we analyze a version of our model in which investments to acquire new products do not scale up one to one with firm size, but instead scale up with the number of products that the firm has (this specification of our model is nested in the generalized model of Akcigit and Kerr 2018). We discuss the extent to which we can extend our analytical results and measurement procedure to this alternative model.

Finally, consider our assumption that all incumbent firms share the same technologies for innovative investment. Lentz and Mortensen (2008), Lentz and Mortensen (2016), and Luttmer (2011) present examples of a Neo-Schumpeterian model (in the first two papers) and an expanding varieties model (in the third paper) in which different types of incumbent firms have different technologies for innovating. To nest these models in our framework, we would index the parameter

η_{m}

and the functions

h (\cdot)

and

ζ (\cdot)

characterizing the investment technologies for incumbent firms to acquire new products and to improve their own products respectively by a firm "type"

i

. The evolution of aggregate productivity in these models is then a function of the measure of intermediate goods produced by firms of each type together with the aggregate investments

x_{m}^{i}

and

x_{c}^{i}

of incumbent firms of each type. To compute the transition dynamics of aggregate productivity implied by these models, one would have to keep track of the evolution of the distribution of intermediate goods produced by each type of firm.

E Variations of baseline model

E. 1 Occupation choice

Suppose that workers each period draw a productivity

a

to work in the research sector, where

a

is drawn from a cumulative distribution function

F (a)

that is Pareto with minimum 1 and slope coefficient

χ > 1

. There are two wages,

W_{p t}

and

W_{r t}

. For the marginal agent,

{\bar{a}}_{t} W_{r t} = W_{p t}

Given that the minimum value of

a

is 1 , any interior equilibrium with positive production requires

W_{r t} \leq W_{p t}

. The aggregate supplies of production and research labor (relative to the total labor force, which grows exogenously) are

l_{p t} = F ({\bar{a}}_{t}) = 1 - {\bar{a}}_{t}^{- χ}

l_{r t} = \int_{{\bar{a}}_{t}}^{\infty} a f (a) d a = \frac{χ}{χ - 1} {\bar{a}}_{t}^{1 - χ}

The ratio of labor

l_{r t} / l_{p t}

and the ratio of wages

W_{r t} / W_{p t}

are determined by

\begin{matrix} \frac{W_{r t} l_{r t}}{W_{p t} l_{p t}} = \frac{μ}{(1 - α)} i_{r t} \\ \frac{l_{r t}}{l_{p t}} = \frac{χ}{χ - 1} \frac{{(\frac{W_{p t}}{W_{r t}})}^{1 - χ}}{1 - {(\frac{W_{p t}}{W_{r t}})}^{- χ}} \end{matrix}

These two equations replace equation (13) in our baseline model to solve for

l_{r t} / l_{p t}

as a function of

i_{r t}

. Note that as

χ

goes to infinity,

W_{r t} / W_{p t}

must converge to 1 in order for

l_{r t} / l_{p t}

to be finite. The elasticity of

l_{r t}

with respect to

i_{r t}

(\log l_{r t}^{'} - \log {\bar{l}}_{r}) = \frac{(χ - 1)}{(χ - 1) (1 + \frac{{\bar{W}}_{r} {\bar{l}}_{r}}{{\bar{W}}_{p} {\bar{l}}_{p}}) + 1} (\log i_{r t}^{'} - \log {\bar{i}}_{r})

When

χ

converges to 1 (high worker heterogeneity), the elasticity of

l_{r t}

with respect to

i_{r t}

converges to 0 . When

χ

converges to infinity (no worker heterogeneity), this elasticity converges to

{\bar{l}}_{p}

, as in our baseline model.

E. 2 Goods and labor used as inputs in research

We consider an extension in which research production uses both labor and the consumption good, as in the lab-equipment model of Rivera-Batiz and Romer (1991), and discuss the central changes to our analytic results. Specifically, the production of the research good is given by

\begin{matrix} (106) & Y_{r t} = A_{r t} Z_{t}^{ϕ - 1} L_{r t}^{λ} X_{t}^{1 - λ} \end{matrix}

and the resource constraint of the final consumption good is

C_{t} + K_{t + 1} - (1 - d_{k}) K_{t} + X_{t} = Y_{t}

Given this production technology, the BGP growth rate of aggregate productivity is given by

{\bar{g}}_{Z} = \frac{{\bar{δ}}_{A_{r}} + {\overset{―}{&}}_{L}}{1 - \tilde{ϕ}}

, where

\tilde{ϕ} = ϕ + \frac{1 - λ}{1 - α}

. The condition for semi-endogenous growth is

\tilde{ϕ} < 1

(equality for endogenous growth).

Revenues from the production of the research good are divided as follows:

\begin{matrix} (107) & W_{t} L_{r t} = λ P_{r t} Y_{r t}, and X_{t} = (1 - λ) P_{r t} Y_{r t} \end{matrix}

The analog to equation (13), relating the allocation of labor between production and research to the innovation intensity, is

\begin{matrix} (108) & l_{r t} = \frac{i_{r t}}{i_{r t} + \frac{(1 - α)}{μ λ}} and l_{p t} = 1 - l_{r t} \end{matrix}

where

i_{r t} \equiv P_{r t} Y_{r t} / ((1 + τ_{y}) Y_{t})

. Our analytical results need to be modified for two reasons. First, for a given share of production labor in output

(1 - α) / μ

, the elasticity of research labor

l_{r t}

with respect to the innovation intensity of the economy

i_{r t}

is decreasing in

λ

. A higher share of goods in production of the research good (lower

λ

) increases the sensitivity of

l_{r t}

with respect to

i_{r t}

Second, by equations (5), (106), (107), and

R_{k t} = (1 + τ_{y}) \frac{α}{μ} \frac{Y_{t}}{K_{t}}

, we have

Y_{r t} = κ A_{r t} Z_{t}^{\tilde{Φ} - 1} {(\frac{K_{t}}{Y_{t}})}^{\frac{α (1 - λ)}{1 - α}} L_{r t}

where

κ

is a constant, so

\log Y_{r t}^{'} - \log {\bar{Y}}_{r} = (\log l_{r t}^{'} - \log {\bar{l}}_{r}) - (1 - \tilde{ϕ}) (\log Z_{t}^{'} - \log {\bar{Z}}_{t}) - \frac{(1 - λ) α}{1 - α} (\log R_{k t}^{'} - \log {\bar{R}}_{k})

.
Long-run changes in aggregate productivity are given as in equation (14) in Proposition 1 where

\tilde{ϕ}

replaces

ϕ

. A lower value of

λ

increases

\tilde{ϕ}

and the associated long-term productivity gains from a given increase in research labor. Following the same steps as in the proof of Proposition 2, we obtain the analog of equation (17) for the new path for aggregate productivity, up to a first-order approximation:

\log Z_{t + 1}^{'} - \log {\bar{Z}}_{t + 1} \approx \sum_{j = 0}^{t} [Γ_{j} (\log l_{r t - j}^{'} - \log {\bar{l}}_{r}) - \frac{(1 - λ) α}{1 - α} (\log R_{k j}^{'} - \log {\bar{R}}_{k})]

where the decay coefficients are given as in equation (18) with

\tilde{ϕ}

replacing

ϕ

. The second term on the right-hand side reflects the change in research output

Y_{r t}

that results when

λ < 1

from changes in the capital-output ratio. This second term is equal to zero once the economy converges to the new long-run BGP. The dynamics of aggregate output are given as in Corollary 1.

E. 3 Klette and Kortum (2004) specification

Here we briefly describe our analytic results in a version of our model that follows Klette and Kortum (2004) more closely in assuming a unitary elasticity of substitution between intermediate goods,

ρ = 1

. With

ρ = 1

, we abstract from growth in the measure of intermediate goods and from innovation by incumbents to improve their own goods.

Specifically, every product that is new to an incumbent firm or an entrant firm is stolen from another incumbent firm (that is,

δ_{m} = δ_{e} = 1

). We normalize the constant total measure of products to

1, M_{t} = 1

. Output of the final good is

\begin{matrix} (109) & \log Y_{t} = \sum_{z} \log (y_{t} (z)) M_{t} (z) \end{matrix}

For each intermediate good, the markup is fixed at

μ

(determined by the productivity distance between the incumbent producer and a latent competitor). The size of each product (in terms of revenues or input use relative to the total) is

s_{t} (z) = 1 / M_{t} = 1

. Substituting the production function (3) into equation (109), and given that the ratio of physical capital to production labor is independent of

z

, aggregate output in equilibrium is

Y_{t} = Z_{t} {(K_{t})}^{α} {(L_{p t})}^{1 - α}

where

Z_{t} = \exp (\sum_{z} z M_{t} (z))

With constant markups, variable profits of each product are given by

\frac{μ - 1}{μ} (1 + τ_{y}) Y_{t}

. Given that variable profits are independent of

z

, firms have no incentive to use the technology

ζ

(.) to invest in improvements of the products they own. Without loss of generality, we set

x_{c t} = 0

and

ζ (0) = 1

An incumbent firm that owns the right to produce a product with productivity

z

possesses the technology to acquire new goods by investing

x_{m t} (z)

units of the research good to displace with probability

h (x_{m t} (z))

a product drawn at random from the entire distribution and start producing at

t + 1

with a productivity index

z^{'}

that is a step

Δ_{s} > 0

higher than the stolen product.

^{32}

Similarly, entrants can invest 1 unit of the research good

to displace and improve by

Δ_{S}

a product drawn randomly from the entire distribution.

^{33}

We conjecture (and verify below) that incumbents' investments are independent of

z

, so

x_{m t} (z) = x_{m t}

. A firm that owns the rights to produce

n

products at time

t

invests a total of

x_{m t} \times n

units of the research to acquire in expectation

h (x_{m t}) \times n

new products (the expectation of a binomial distribution with parameters

h (x_{m t})

and

n

). Note that this innovative technology for incumbents can be equivalently described (as in Klette and Kortum 2004) as an investment of

c (I_{t} / n) \times n

to acquire

I_{t} \times n

products in expectation, where

c (\cdot)

is increasing and convex. To map these two technologies, we set

I_{t} = h (x_{m t})

and

c (I_{t} / n) = x_{m t}

The measure of incumbent products that are displaced and improved is

h (x_{m t}) \times 1 +

x_{e t}

, and the

G

function mapping aggregate investment levels to productivity growth is

G (x_{m t}, x_{e t}) = (h (x_{m t}) + x_{e t}) Δ_{s}

Impact elasticities are

\begin{matrix} Θ_{m} \equiv G_{m} ({\bar{x}}_{m}, {\bar{x}}_{e}) {\bar{Y}}_{r} = h^{'} ({\bar{x}}_{m}) Δ_{s} {\bar{Y}}_{r} \\ Θ_{e} \equiv G_{e} ({\bar{x}}_{m}, {\bar{x}}_{e}) {\bar{Y}}_{r} = Δ_{s} {\bar{Y}}_{r} \end{matrix}

Note that

Θ_{e}

can also be written as

\begin{matrix} (110) & Θ_{e} = Δ_{s} {\bar{x}}_{e} {\bar{Y}}_{r} = ({\bar{g}}_{Z} - G ({\bar{x}}_{m}, 0)) \frac{{\bar{Y}}_{r}}{{\bar{x}}_{e}} \end{matrix}

which corresponds to the upper bound in equation (37). If the equilibrium allocation of innovative investment is conditionally efficient (i.e.,

h^{'} ({\bar{x}}_{m}) = 1

, so that

Θ_{m} = Θ_{e}

), then the impact elasticity

Θ

(for any perturbation to

x_{m t}

and

x_{e t}

) has an upper bound as given in equation (26) with

G (0) = 0

. Hence,

Θ \leq {\bar{g}}_{Z}

The value of a firm that owns the rights to produce

n

products (independently of their productivity) is equal to

V_{t} n

, with

\begin{matrix} (111) & V_{t} = max_{x_{m} \geq 0} \frac{μ - 1}{μ} (1 + τ_{y}) Y_{t} - (1 - τ_{m t}) P_{r t} x_{m} + \frac{1}{1 + R_{t}} (1 - δ_{c t} + h (x_{m})) V_{t + 1} \end{matrix}

where the displacement probability of each product,

δ_{c t}

, is taken as given by the firm and, in equilibrium, is equal to

δ_{c t} = h (x_{m t}) + x_{e t}

. The first-order condition of this profit

maximization implies that

x_{m t} (z) = x_{m t}

with

\begin{matrix} (112) & (1 - τ_{m t}) P_{r t} \geq \frac{1}{1 + R_{t}} h^{'} (x_{m t}) V_{t + 1} \end{matrix}

confirming that

x_{m t}

is independent of

z

. The free entry condition is

\begin{matrix} (113) & (1 - τ_{e t}) P_{r t} \geq \frac{1}{1 + R_{t}} V_{t + 1} \end{matrix}

The equilibrium allocation of innovative investment is conditionally efficient

(h^{'} ({\bar{x}}_{m}) =

1) if and only if innovation policies are uniform

(τ_{m t} = τ_{e t})

Finally, we show how we can measure

\frac{{\bar{P}}_{r} {\bar{x}}_{e}}{(1 + τ_{y}) {\bar{Y}}_{t}}

, which is required to calculate

Θ_{e}

in equation (110). Defining

v_{t} = V_{t} / ((1 + τ_{y}) Y_{t})

and using the fact that, with

ρ = 1

and

g_{M t} = 0, s_{e t + 1} = f_{e t + 1} = x_{e t}

, we have from equation (113),

where by equation (111),

\bar{v} = \frac{[\frac{μ - 1}{μ} - (1 - {\bar{τ}}_{m}) \frac{{\bar{P}}_{r} {\bar{x}}_{m}}{(1 + τ_{y}) {\bar{Y}}_{t}}]}{1 - \frac{\exp ({\bar{g}}_{Y})}{1 + \bar{R}} (1 - {\bar{s}}_{e})}

which is analogous to equation (90) in the baseline model.

E. 4 Alternative specification of incumbent technology to acquire new products

In this section, we consider a specification of the investment technology for incumbent firms to acquire new products (which is nested in the generalized model of Akcigit and Kerr 2018) in which investments to acquire new products do not scale up one to one with firm size, but instead scale up with the number of products that the firm has. These two specifications are equivalent when

ρ = 1

, as discussed in Section E.3.

We assume that a firm with a product with productivity

z

has a technology to acquire a new product with probability

h (x_{m t} (z) M_{t})

(as opposed to

h (x_{m t} (z) / s_{t} (z))

in our baseline model). With probability

δ_{m}

it displaces a product drawn at random from the entire distribution (as opposed to displacing a product with productivity

z

), so that the displaced product has productivity index

z

at time

t

with

E z^{ρ - 1} = Z_{t}^{ρ - 1} / M_{t}

. The
incumbent firm that stole this product can produce it at

t + 1

with a new productivity index

z^{'}

such that the expected value of the term

z^{' ρ - 1}

is equal to

E z^{' ρ - 1} = η_{m s} Z_{t}^{ρ - 1} / M_{t}

(instead of

η_{m s} z^{ρ - 1}

in the baseline model), with

η_{m s} > 1

. With complementary probability

1 - δ_{m}

the newly acquired product is new to society, with productivity index

z^{'}

drawn from a distribution such that the expected value of the term

z^{ρ ρ - 1}

is equal to

E z^{' ρ - 1} = η_{m n} Z_{t}^{ρ - 1} / M_{t}

(instead of

η_{m n} z^{ρ - 1}

in the baseline model), with

η_{m n} > 0

. We define

η_{m} = δ_{m} η_{m s} + (1 - δ_{m}) η_{m n}

. The investment technologies of entrants and of incumbent firms to improve continuing products remain unchanged.

We conjecture (and verify below) that the level of investment by incumbent firms in acquiring new products is independent of the productivity index

z

of the product with which this investment is associated. So,

x_{m t} (z) = x_{m t} / M_{t}

(instead of

x_{m t} (z) = s_{t} (z) x_{m t}

in the baseline model). We also conjecture that investments by incumbent firms in improving their continuing products scale up with the size of the product with which this investment is associated, as in the baseline model. With

x_{m t} (z) = x_{m t} / M_{t}

and

x_{c t} (z) =

x_{c t} s_{t} (z)

, a firm that owns the right to produce

n

products with productivities

{z_{1}, \dots, z_{n}}

has size

s_{t} = \sum_{i = 1}^{n} s_{t} (z_{i})

and spends a total

r_{t} = s_{t} (x_{c t} + \frac{x_{m t}}{M} s_{t})

on the research good. Assuming a positive cross-firm correlation between firm size

s_{t}

and average product size

s_{t} / n

, this alternative specification generates a negative correlation between firm size

s_{t}

and innovation intensity

\frac{r_{t}}{s_{t}} = x_{c t} + \frac{x_{m t}}{M} \frac{n}{s_{t}}

. Similarly, since small firms have products with a lower average

z

, their investments in acquiring new products produce mean reversion in the average value of

z

(since the new

z^{'}

s they acquire are drawn from the entire distribution and not from products with productivity

z

as in the baseline model), implying higher growth relative to firms with a higher average value of

z .^{34}

In what follows, we discuss how our analytic results are affected under the modified technology.

Incumbent firms acquire in the aggregate a measure

h (x_{m t}) M_{t}

new products - a measure

δ_{m} h (x_{m t}) M_{t}

are stolen from other incumbent firms and complementary measure

(1 - δ_{m}) h (x_{m t}) M_{t}

are new to society. The average value of

z^{' ρ - 1}

across all newly acquired products in incumbent firms at

t + 1

η_{m} \frac{Z_{t}^{ρ - 1}}{M_{t}}

. With

x_{c t} (z) = s_{t} (z) x_{c t}

, and following the logic of the proof of Lemma 3, we obtain the same law of motion for aggregate productivity (28) and resource constraint of the research good (27) as in our baseline model. Therefore,

Θ_{c}, Θ_{m}

and

Θ_{e}

are given as in equations (66)-(68) as well as equation (36) in

our baseline model.
We now verify our conjecture that in equilibrium, firms choose

x_{m t} (z) = x_{m t} / M_{t}

and

x_{c t} (z) = s_{t} (z) x_{c t}

. Under this conjecture, the value of a continuing product with productivity index

z

for an incumbent firm is given by

V_{t} (z) + U_{t}

where

V_{t} (z)

denotes the discounted present value for an incumbent firm of the dividends associated with a product with productivity

z

at time

t

, and

U_{t}

denotes the value for an incumbent firm of this product in facilitating further acquisition of products (which, in contrast to our baseline model, is independent of

z

). The value of a firm that owns the technology to produce

n

products with productivities

{z_{1}, \dots, z_{n}}

is equal to

V_{t} (z) \sum_{i = 1}^{n} s_{t} (z_{i}) + U_{t} n

Specifically,

V_{t} (z) = s_{t} (z) V_{t}

with

\begin{matrix} (114) & V_{t} = max_{x_{c} \geq 0} (1 + τ_{y}) \frac{μ - 1}{μ} Y_{t} - (1 - τ_{c t}) P_{r t} x_{c} + \frac{1}{1 + R_{t}} V_{t + 1} (1 - δ_{c t}) ζ (x_{c}) \frac{Z_{t}^{ρ - 1}}{Z_{t + 1}^{ρ - 1}} \end{matrix}

and

U_{t} = max_{x_{m} \geq 0} - (1 - τ_{m t}) P_{r t} x_{m} + \frac{1}{1 + R_{t}} (h (x_{m} M_{t}) V_{t + 1} η_{m} \frac{Z_{t}^{ρ - 1}}{M_{t} Z_{t + 1}^{ρ - 1}} + (1 - δ_{c t} + h (x_{m} M_{t})) U_{t + 1})

The first-order condition of the incumbent firm's profit maximization problem (114) with respect to

x_{c}

implies that

x_{c t} (z) = s_{t} (z) x_{c t}

with

\begin{matrix} (116) & (1 - τ_{c t}) P_{r t} \geq \frac{1}{1 + R_{t}} V_{t + 1} (1 - δ_{c t}) ζ^{'} (x_{c t}) \frac{Z_{t}^{ρ - 1}}{Z_{t + 1}^{ρ - 1}} \end{matrix}

and the first-order condition of the incumbent firm's profit maximization problem (115) with respect to

x_{m}

implies that

x_{m t} (z) = x_{m t} / M_{t}

with

\begin{matrix} (117) & (1 - τ_{m t}) P_{r t} \geq \frac{1}{1 + R_{t}} h^{'} (x_{m t}) (V_{t + 1} η_{m} \frac{Z_{t}^{ρ - 1}}{Z_{t + 1}^{ρ - 1}} + U_{t + 1} M_{t}) \end{matrix}

The free entry condition is

\begin{matrix} (118) & (1 - τ_{e t}) P_{r t} \geq \frac{1}{1 + R_{t}} (V_{t + 1} η_{e} \frac{Z_{t}^{ρ - 1}}{Z_{t + 1}^{ρ - 1}} + U_{t + 1} M_{t}) \end{matrix}

These first-order conditions are equalities if

x_{c t}, x_{m t} > 0

.
Defining

v_{t} = V_{t} / ((1 + τ_{y}) Y_{t})

and

u_{t} = U_{t} M_{t} / ((1 + τ_{y}) Y_{t})

, and assuming an interior equilibrium, we can rewrite this system of equations as

\begin{matrix} (119) & \frac{(1 - τ_{c t}) P_{r t}}{(1 - τ_{y}) Y_{t}} = \frac{Y_{t + 1} / Y_{t}}{1 + R_{t}} v_{t + 1} (1 - δ_{c t}) ζ^{'} (x_{c t}) \frac{Z_{t}^{ρ - 1}}{Z_{t + 1}^{ρ - 1}} \\ (120) & \frac{(1 - τ_{m t}) P_{r t}}{(1 - τ_{y}) Y_{t}} = \frac{Y_{t + 1} / Y_{t}}{1 + R_{t}} h^{'} (x_{m t}) (v_{t + 1} η_{m} \frac{Z_{t}^{ρ - 1}}{Z_{t + 1}^{ρ - 1}} + u_{t + 1} \frac{M_{t}}{M_{t + 1}}) \\ (121) & \frac{(1 - τ_{e t}) P_{r t}}{(1 - τ_{y}) Y_{t}} = \frac{Y_{t + 1} / Y_{t}}{1 + R_{t}} (v_{t + 1} η_{e} \frac{Z_{t}^{ρ - 1}}{Z_{t + 1}^{ρ - 1}} + u_{t + 1} \frac{M_{t}}{M_{t + 1}}) \\ v_{t} = \frac{μ - 1}{μ} - \frac{(1 - τ_{c t}) P_{r t} x_{c t}}{(1 - τ_{y}) Y_{t}} + \frac{Y_{t + 1} / Y_{t}}{1 + R_{t}} v_{t + 1} (1 - δ_{c t}) ζ (x_{c t}) \frac{Z_{t}^{ρ - 1}}{Z_{t + 1}^{ρ - 1}} \\ u_{t} = - \frac{(1 - τ_{m t}) P_{r t} x_{m t}}{(1 + τ_{y}) Y_{t}} + \frac{Y_{t + 1} / Y_{t}}{1 + R_{t}} (h (x_{m t}) v_{t + 1} η_{m} \frac{Z_{t}^{ρ - 1}}{Z_{t + 1}^{ρ - 1}} + (1 - δ_{c t} + h (x_{m t})) u_{t + 1} \frac{M_{t}}{M_{t + 1}}) \end{matrix}

Recall that in the baseline model, if new innovation policies satisfy condition (32), for a given value of

Y_{r t}

we are able to solve for

x_{c t}, x_{m t}

, and

x_{e t}

as a static system of equations using equations (64), (65), and the resource constraint (27). In contrast, in this alternative model specification, we must also solve for

v_{t + 1}

and

u_{t + 1}

, unless

x_{c t}

is fixed exogenously

(x_{c t} = {\bar{x}}_{c})

and

η_{m} = η_{e}

. In this case,

x_{m t}

and

x_{e t}

can be solved using the two following static equations:

\frac{1 - τ_{m t}}{1 - τ_{e t}} = h^{'} (x_{m t}) and {\bar{x}}_{c} + x_{m t} + x_{e t} = Y_{r t}

without requiring to solve for

v_{t + 1}

and

u_{t + 1}

.
In the BGP,

x_{c t} = {\bar{x}}_{c}, x_{m t} = {\bar{x}}_{m}, x_{e t} = {\bar{x}}_{e}, v_{t} = \bar{v}

, and

u_{t} = \bar{u}

. The first-order conditions (119), (121) in the BGP are

\begin{matrix} (122) & \frac{(1 - τ_{c}) {\bar{P}}_{r t}}{(1 + τ_{y}) {\bar{Y}}_{t}} = \frac{\exp ({\bar{g}}_{Y} - (ρ - 1) {\bar{g}}_{Z})}{1 + \bar{R}} \bar{v} (1 - {\bar{δ}}_{c}) ζ^{'} ({\bar{x}}_{c}) \\ (123) & \frac{(1 - τ_{m}) {\bar{P}}_{r t}}{(1 + τ_{y}) {\bar{Y}}_{t}} = \frac{\exp ({\bar{g}}_{Y} - (ρ - 1) {\bar{g}}_{Z})}{1 + \bar{R}} h^{'} ({\bar{x}}_{m}) (\bar{v} η_{m} + \bar{u} \exp ((ρ - 1) {\bar{g}}_{Z} - {\bar{g}}_{M})) \\ (124) & \frac{(1 - τ_{e}) {\bar{P}}_{r t}}{(1 + τ_{y}) {\bar{Y}}_{t}} = \frac{\exp ({\bar{g}}_{Y} - (ρ - 1) {\bar{g}}_{Z})}{1 + \bar{R}} (\bar{v} η_{e} + \bar{u} \exp ((ρ - 1) {\bar{g}}_{Z} - {\bar{g}}_{M})) \end{matrix}

where

\bar{v}

and

\bar{u}

satisfy

\begin{matrix} (125) & \bar{v} = \frac{μ - 1}{μ} - \frac{(1 - τ_{c}) {\bar{P}}_{r t}}{(1 + τ_{y}) {\bar{Y}}_{t}} + \frac{\exp ({\bar{g}}_{Y} - (ρ - 1) {\bar{g}}_{Z})}{1 + \bar{R}} \bar{v} (1 - {\bar{δ}}_{c}) ζ ({\bar{x}}_{c}) \\ (126) & \bar{u} = - \frac{(1 - τ_{m}) {\bar{P}}_{r t}}{(1 + τ_{y}) {\bar{Y}}_{t}} {\bar{x}}_{m} + \frac{\exp ({\bar{g}}_{Y} - (ρ - 1) {\bar{g}}_{Z})}{1 + \bar{R}} (h (x_{m t}) η_{m} \bar{v} + (1 - {\bar{δ}}_{c} + h ({\bar{x}}_{m})) \bar{u} \exp ((ρ - 1) {\bar{g}}_{Z} - {\bar{g}}_{M})) \end{matrix}

In equilibrium, it must be that

\bar{v} \geq 0

and

\bar{u} \geq 0

.
Using equations (122), (123), and (124),

Θ_{c}, Θ_{m}

and

Θ_{e}

can be expressed as

\begin{matrix} Θ_{c} = [\frac{1 - τ_{c}}{ρ - 1}] [\frac{1 + \bar{R}}{\exp ({\bar{g}}_{Y})}] \frac{{\bar{i}}_{r}}{\bar{v}} \\ Θ_{m} = [\frac{1 - τ_{m}}{ρ - 1}] [1 - δ_{m} \frac{ζ ({\bar{x}}_{c})}{η_{m}}] [\frac{1 + \bar{R}}{\exp ({\bar{g}}_{Y})}] \frac{{\bar{i}}_{r}}{\bar{v}} \frac{1}{[1 + \frac{\bar{u}}{\bar{v} η_{m}} \exp ((ρ - 1) {\bar{g}}_{Z} - {\bar{g}}_{M})]} \\ Θ_{e} = [\frac{1 - τ_{e}}{ρ - 1}] [1 - δ_{e} \frac{ζ ({\bar{x}}_{c})}{η_{e}}] [\frac{1 + \bar{R}}{\exp ({\bar{g}}_{Y})}] \frac{{\bar{i}}_{r}}{\bar{v}} \frac{1}{[1 + \frac{\bar{u}}{\bar{v} η_{e}} \exp ((ρ - 1) {\bar{g}}_{Z} - {\bar{g}}_{M})]} \end{matrix}

where

\frac{ζ ({\bar{x}}_{c})}{η_{m}} = \frac{a v \overset{―}{s i z e}}{a v \bar{z} i z e_{m}}, \frac{ζ ({\bar{x}}_{c})}{η_{e}} = \frac{a v \bar{s} i z e_{c}}{a v \bar{s} i z e_{e}}, η_{m} \exp ({\bar{g}}_{M} - (ρ - 1) {\bar{g}}_{Z}) = {avsize}_{m}

and

η_{e} \exp ({\bar{g}}_{M} -

(ρ - 1) {\bar{g}}_{Z}) = a_{v} {\overset{―}{s i z e}}_{e}

as in our baseline model. These expressions coincide with expressions

86, (87)

and (88) in our baseline model with the exception of the added terms

{[1 + \frac{\bar{u}}{\bar{v} \times a v \bar{s} i z e_{m}}]}^{- 1}

and

{[1 + \frac{\bar{u}}{\bar{v} \times a v \bar{s} i z e_{e}}]}^{- 1}

Θ_{m}

and

Θ_{e}

. The ranking of impact elasticities is equal to that in our baseline specification. Lemma 4 providing conditions under which the equilibrium allocation of innovative investment on the initial BGP is conditionally efficient holds, except that in case (i) there is an additional requirement that

\overset{―}{\bar{u}} \to 0

We now describe how we infer the value of

\frac{{\bar{P}}_{r} + {\bar{x}}_{e}}{{\bar{Y}}_{t}}

, which we use to measure

Θ_{e}

in equation (36) (and which is also used to measure the remaining impact elasticities). By the free entry condition (124),

\begin{matrix} (127) & \frac{{\bar{P}}_{r t} {\bar{x}}_{e}}{(1 + τ_{y}) {\bar{Y}}_{t}} = \frac{\exp (\bar{g} Y)}{1 + \bar{R}} \frac{\bar{v} {\bar{s}}_{e}}{(1 - τ_{e})} (1 + \frac{\bar{u}}{\bar{v} \times a v \bar{s} i z e_{e}}) \end{matrix}

where

{\bar{s}}_{e}

and

a v \bar{s} {\overset{―}{s i z e}}_{e}

can be measured as described in Appendix C.3. Combining equations (125) and (126), we obtain

\begin{matrix} (128) & \bar{v} (1 + \frac{\bar{u}}{\bar{v}}) = \frac{μ - 1}{μ} - \frac{(1 - τ_{c}) {\bar{P}}_{r t} ({\bar{x}}_{c} + {\bar{x}}_{m})}{(1 + τ_{y}) {\bar{Y}}_{t}} + \frac{\exp (\bar{g} Y)}{1 + \bar{R}} \bar{v} (1 - {\bar{s}}_{e}) [1 + \frac{\bar{u}}{\bar{v} \times {avsize}_{c m}}] \end{matrix}

where

{avsize}_{c m} = \frac{(1 - {\bar{s}}_{e})}{(1 - {\bar{f}}_{e})}

Given a value of

\bar{u} / \bar{v}

and measures of

{\bar{s}}_{e}

a v \bar{s} i z e_{c m}

, and

\frac{{\bar{P}}_{r t} ({\bar{x}}_{c} + {\bar{x}}_{m})}{(1 + τ_{y})}

, we use equation (127) to calculate

\bar{v}

and

\frac{{\bar{P}}_{r} {\bar{x}}_{e}}{(1 + τ_{y}) {\bar{Y}}_{t}}

, with which we can then calculate all impact elasticities.

While the ratio

\bar{u} / \bar{v}

cannot be inferred directly, we can bound it as follows. Equation (125) provides an upper bound for

\bar{v} / (\bar{v} + \bar{u})

(since

{\bar{x}}_{c} \geq 0

), and equation (126) provides a lower bound for

\bar{v} / (\bar{v} + \bar{u})

(since

{\bar{x}}_{m} \geq 0)

. Given

\bar{v} / (\bar{v} + \bar{u})

, we can pin down the value of

{\bar{x}}_{c}

relative to

{\bar{x}}_{m}

Finally, note that using equations (124), (125), and (126), the total value of incumbent firms relative to output in the BGP is given by

\bar{v} + \bar{u} = \frac{1}{(1 - \frac{\exp (\bar{g} r)}{1 + \bar{R}})} [\frac{μ - 1}{μ} - \frac{{\bar{P}}_{r t}}{(1 + τ_{y}) {\bar{Y}}_{t}} ((1 - τ_{c}) {\bar{x}}_{c} + (1 - τ_{m}) {\bar{x}}_{m} + (1 - τ_{e}) {\bar{x}}_{e})]

which coincides with the expression (91) for

\bar{v}

in our baseline model.

References

Akcigit, Ufuk and William R. Kerr, "Growth through Heterogeneous Innovations," Journal of Political Economy, 2018, 126 (4), 1374-1443.

Garcia-Macia, Daniel, Pete Klenow, and Chang-Tai Hsieh, "How Destructive is Innovation?," Working Paper 22953, National Bureau of Economic Research, December 2016.

Hall, Robert E., "Corporate Earnings Track the Competitive Benchmark," Working Paper 10150, National Bureau of Economic Research, November 2003.

Klette, Tor Jakob and Samuel Kortum, "Innovating Firms and Aggregate Innovation," Journal of Political Economy, October 2004, 112 (5), 986-1018.

Lentz, Rasmus and Dale T. Mortensen, "An Empirical Model of Growth Through Product Innovation," Econometrica, November 2008, 76 (6), 1317-1373.
_ and _ , "Optimal Growth Through Product Innovation," Review of Economic Dynamics, January 2016, 19, 4-19.

Luttmer, Erzo G. J., "On the Mechanics of Firm Growth," Review of Economic Studies, 2011, 78 (3), 1042-1068.

McGrattan, Ellen R. and Edward C. Prescott, "Taxes, Regulations, and the Value of U.S. and U.K. Corporations," Review of Economic Studies, July 2005, 72 (3), 767-796.

Peters, Michael, "Heterogeneous Markups, Growth and Endogenous Misallocation," Unpublished Paper, Yale University 2016.

Poterba, James M., "The Rate of Return to Corporate Capital and Factor Shares: New Estimates Using Revised National Income Accounts and Capital Stock Data," CarnegieRochester Conference Series on Public Policy, June 1998, 48, 211-246.

Rivera-Batiz, Luis A. and Paul M. Romer, "Economic Integration and Endogenous Growth," Quarterly Journal of Economics, May 1991, 106 (2), 531-555.

Tyson, Laura and Greg Linden, "The Corporate R&D Tax Credit and U.S. Innovation and Competitiveness; Gauging the Economic and Fiscal Effectiveness of the Credit," Working Paper, Center for American Progress, January 6, 2012.

*UCLA and Federal Reserve Bank of Minneapolis
$^{†}$ UCLA
$^{27}$ The agents' intertemporal elasticity of substitution is used only in computing the transition dynamics of physical capital, for given changes in research labor as we do in our policy counterfactuals. Hence, our results regarding the transition dynamics of aggregate productivity do not depend on this parameter. Moreover, our results on welfare, to a first approximation, do not depend on this parameter. Only the transition dynamics of aggregate output conditional on the transition dynamics of aggregate productivity are affected by the intertemporal elasticity of substitution.
$^{28}$ This value of production subsidy is set so that $(1 - τ_{corp}) ((1 + τ_{y}) \frac{α y}{μ} \overset{―}{\bar{k}} - d_{k}) = α \frac{y}{\bar{k}} - d_{k} = \bar{R}$ .
$^{29}$ If we set $ζ_{2} = 0.05$ , close to its lower bound, then $Θ = 0.0098$ .
$^{30}$ The Matlab codes that we use to solve the model numerically (linearly and nonlinearly) are available at www.econ.ucla.edu/arielb/innovationpolicycodes.zip.
$^{32}$ Klette and Kortum (2004) consider an extension of their model in which firms differ permanently in terms of the size of their step sizes (and markups) on the products they own. With $ρ = 1$ , markup heterogeneity does not affect aggregate productivity. Since the cost of innovation also scales up with step size (or equivalently, the probability of success falls with step size), all firms choose the same innovation investment per product.
$^{33}$ In our baseline model, we have assumed that innovations associated with product $z$ displace other products with productivity z, whereas here (as well as in Klette and Kortum 2004) we have assumed that displaced products are drawn randomly from the entire distribution (see also Appendix E.4). When $ρ = 1$ , these two assumptions have identical aggregate implications since size is independent of $z$ .
$^{34}$ To match the quantitative magnitude of these effects, Akcigit and Kerr (2018) argue that investments by incumbent firms to acquire new products scale moderately slower than the number of products in the firm. As a result, the relationship between aggregate productivity growth and aggregate innovative investment in their estimated model is a function of the firm size distribution. Thus, in order to assess the impact of innovation policies on the dynamics of aggregate productivity, one would have to solve their model fully numerically.